Partial regularity and extension of solutions to the Navier-Stokes equations (Mathematical Analysis of Viscous Incompressible Fluid)

Abstract

It is quite well-known that we cannot assure the existence of global-in-time solutions to the Navier-Stokes equations for large initial data, but we have local-in-time solutions at least. The purpose of this talk is to get another time extension criterion for that local-intime solution. Specifically, We work on smooth classical solutions which satisfy so called Leray-Hopf class on mathbb{R}^{n}times(0, T), and then establish an time-extension criterion beyond T by estimating a sort of Morrey type functional of solutions. A key idea here is to utilize the $epsilon$-regularity argument which has been the critical part of the theory of suitable weak solutions. We note that this article is based on the author s recent work [23] and also contains similar results for bounded domains.